NCERT Class 12 Maths · Chapter 4 Determinants · Exercise 4.5 Question 1

✅ Step-by-Step Solution

1. For a 2×2 matrix use det A=a₁₁a₂₂−a₁₂a₂₁.
2. \(3\) − \(4\) = \(-1\).
3. First check whether this determinant is zero.
4. The cofactor matrix is \(\left[\begin{matrix}3 & -2\\-2 & 1\end{matrix}\right]\).
5. Transpose the cofactor matrix: adj A=\(\left[\begin{matrix}3 & -2\\-2 & 1\end{matrix}\right]\).
6. Divide by det A: A⁻¹=\(\left[\begin{matrix}-3 & 2\\2 & -1\end{matrix}\right]\).
7. Multiplication verifies AA⁻¹=A⁻¹A=I.
8. Multiply X=A⁻¹B=\(\left[\begin{matrix}0\\1\end{matrix}\right]\).
9. Substituting this vector into the original equations gives AX=\(\left[\begin{matrix}2\\3\end{matrix}\right]\)=B. Hence the system has a unique solution.